Post on 04-Jan-2016
Effect of thermal fluctuation of baryons on vector mesons and low mass dileptons
ρ
ω
eee
e
Sanyasachi Ghosh (VECC, Kolkata, India)
gluon ofpart kinetic )(,
QCD
qmDiqLduq
q
RL
55
q q
)1(2
1 )1(
2
1
qqq
)}( ) {(,
QCD RqLRqLRRduq
LL qmqqmqqDiqqDiqL Mass term 0
)e( 2/i,,,,
RL, RLRLRLRL UqUq
remain invariant under global symmetry &associated conserved Noether currents
RL SU(2)SU(2)
RLa
RLaRL qqJ ,,, )2/(
)(1 )2/(
)(1 )2/( 5
-
LRaa
A
LRaa
V
JJqqJ
JJqqJ Parity doublet
Vector current
Axial vector current
Real world
Spontaneous Breaking
of Chiral Symmetry
(S
BCS)
Broken Chiral Symmetry Restored Chiral Symmetry
Vacuum Stronglyinteracting matter
The spectral properties of vector mesons and axial Vector mesons in strongly-interacting matter may indicate about Chiral Symmetry Restoration (CSR).
Spectral function of ρ and a_1 are measured via hadronic decays of τ lepton at LEP by ALEPH and OPAL
experimental data from heavy ion collisionseems to prefer melting scenario
Rapp et.al. arXiv:0901.3289
Vector mesons are very special to study its in-medium property. Because they are directly couplewith electromagnetic (e.m.) channels.
e
e
matter) ofdimension (D
D 1
path freeMean
e.m.
Low mass enhancement due to in-medium modificationof light vector mesons (specially ρ).
Being motivated by this in-medium signature,we have evaluated spectral functions of lightVector mesons (ρ and ω) in hot and dense hadronic matter using Real-time formalism ofThermal Field theory (RTF).
Fourier transform
t
0 1 2 3 4 5
N(t)/N(0)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Probabilistic amplitude of Unstable particle
time space (t) energy space
)2/exp(~)(
)exp()0(
)(
)0(
)(2
ttimt
tt
N
tN
H
)(~ Im)( ,2/)(
1~)(~
)()2/()(~)(~
)(
)(
000
0
0
2
00
qqimq
q
qm
q
mN
qN
H
HH
5
)(~)( 000
Lim Hmqq
QFT definition of spectral function
),( xtx
),( 0 qqq
imq
mqq
H
H
22
22
0
1 Im~
)(~)(Lim
0|)()(|0 21 xx Propagator
Spectral function
= + + + ……………∞
=0 + 00 + 000 + ………….(an infinite g.p. series)
=0
0
1
= 22
1
baremq
where is called self energy term
220
1
baremq
) Im() Re(q
1
vacvac22
imbare
Vacuum interacting propagator
)q(q
1)q(
2222
mD
imq
mqq
H
H
22
22
0
1 Im~
)(~)(LimVacuum free propagator
) Im()(q
1
vac22
imphy Im = m
thvactotal
Mass shift
Width enhancement
m)(
ReReRe2
2vac
2
phy
thphythbare
m
mm
Spectral function = Im [propagator]0
e e
]})(q{
1[Im)(
22vdecayphy iqm
qA
]}q{
1[Im)(
tot22tot
barem
qA ]}Im)Re(q{
1[Im)(
tottot22tot
barem
qA ])}()(q{
1[Im)(
2phy
2tot
decayiqmmqA
VacuumMedium
-∞ +∞t-axis - ib
0-∞
(+∞,-ib/2)
+∞
t-plane
i t
) H ( exp) t H i ( exp
Time evolution operator
~ density matrix
Essence of RTF
Field Theory of vacuum Field Theory at finite temperature
dt contour
d
thermal propagator
Thermal self-energy
abD
ab),,q(q
1),,q(
0220
TqmTqD
Diagonalization
H
H
e
eTr
}{ AA
Thermal averaged
0 0 A
Vacuum expectation
)()()()(or )()( 332211 yxyxyx
Self Energy of ρ for mesonic loops :
T)k,-q(pD T)(k,D k)(q,N)2(
),( 11114
4
11
kd
Tq
(k)(q) (q)
(p=q – k)
Thermal vacuum11 D
)((..)),(Im 0,
311 pk
ba
baqkdTq
,, , ,ba,
0Lq
0Lq 0
Uq0Uq 0
Landau cuts Unitary cutsUnitary cuts
220 )( kpL mmqq 220 )( kpU mmqq
})nn(1-)n(1{nvac~Im
pkpk
m
ML
)1170(h 1ρ p p ρ
Mesonic collision rate
k k
)( MC
}nn-)n)(1n{(1vac~Im
pkpk
m
MU
ρ ρk k
pp
Bose enhancement of decay rate )( ..M
eB
)1260(a 1
)782(
N
B
N
B+
2B )m(m qMN
o
Unitary cut
)m(mB N
Self Energy of ρ for baryonic loops :
] }nn-)n)(1n{(1 }nn-)n)(1n{(1 [vac~Im
BNBN-BN
-BN
m
BU
ρ ρ
),( NN ),( NN
),( BB ),( BB
Pauli blocking of decay rate )( ..B
bP
Landau cut
] )}n-(1n-)nn{(1 )}n-(1n-)nn{(1 [vac~Im
BNBNBN-BN
m
BL
ρ
),( BB ),( BB
ρ),( NN ),( NNBaryonic collision rate )( B
C
12 2qM
o
Unitary cutLandau cut
N
BN*(1520)
N*(1650)
N*(1720)
)1232(
)1620(*
)1700(*
----------------------------vacuum part of loop (unitary cut )
-------------------------thermal part of loops (unitary cut)
---------------------------- loops (Landau cut )
----------------------------
loops (Landau cut)
MeV 150decay MeV 25.. M
eB MeV 55MC MeV 20B
C
MeV 55MC
MeV 20BC
MeV 150decay
MeV 25. MeB
) , ,
(
11 ah
H
),,,
(**
NNNNNN
NB
MeV 250 & MeV 150 B T
Effect of baryonic chemical potential on ρ spectral function in low mass region:
),,,( 0BTqqA
Effect of temperature on ρ spectral function in low mass region:
),,,( 0BTqqA
Effect of momentum of ρ in off mass shell on its spectral function in low mass region:
),,,( 0BTqqA
N
B
S. Sarkar & S. GhosharXiv:1204.0893
ρ meson spectral function
ω meson spectral function
S. Ghosh and S. Sarkar arXiv:1207.2251 [nucl-th]
I )(J)( )()(J dFie)(2nd
)exp(
lh442
4int
yyAxAxyxd
xdLiSFI
)( )( )( )( int xAxJexAxJeL lh
Strong interacting current
Leptonic current
Z
)Eexp(
)2()2( N
2
, ,
I3
23
31
3
21
pp FI
FISVpdVpd
Final phase space Thermal average
Dilepton multiplicity
WL
qxdd
dN (...)
44
Formalism ofdilepton :
WL
qd
dR (...)
4
)0(J)(J e ),(W hhiq40 xxdiqq x
ssdduudduuem
3
1 2/)(
3
1 2/)(J
HM ),,,( (...) 0
,,4 B
ii TqqAL
qd
dR
J 3
1 J
3
1 J J
em
ssdduuem
3
1
3
1
3
2J
QM
4qd
dRq
q
l
l
2
Contribution of ω is down by a factor ~ 10
Effect of mesonic as well as baryonic medium modification of ρ on dilepton rate in low mass region :
),,,(A (...) 0422 B
T
TqqLqd
dR
dyqddM
dR
qxdd
d44
0 }T,q,N{q
23
},{ 0 qq
)(xu
)(xT
Dilepton production ininvariant mass after Space time evolution :
xdqxdd
xxud 444
})T(,q),(N{q
])}T(,q),(N{q
[)()(
4424
qxdd
xxudqMdydxd
dM
MdNT
Invariant mass spectra
Fluid element
Low mass enhancement at SPS :
Phys.Rev. C85 (2012) 064906J K Nayak, J Alam, T Hirano, S Sarkar and B Sinha
S. Sarkar & S. GhosharXiv:1204.0893
Meson (S.Ghosh, S.Mallik. S.Sarkar Eur. Phys. C 70, (2010) 251) + Bayon (S.Ghosh, S.Sarkar Nucl. Phys. A 870, (2011) 94) -loop self-energies
Meson loop self-energies (S.Ghosh, S.Mallik. S.Sarkar Eur. Phys. C 70, (2010) 251) + Baryon part from Eltesky et. al. (Phys. Rev. C 64, (2001) 035202)
S. Sarkar & S. GhosharXiv:1204.0893
Analytic structure of ρ meson propagator at finite temperatureEur. Phys. J. C 70, 251 (2010).S.Ghosh, Sourav Sarkar, (VECC) , S. Mallik, (SINP)
ρ self-energy at finite temperature and density in thereal-time formalismNucl. Phys. A 870, 94 (2011) S. Ghosh, Sourav Sarkar, (VECC)
Observing many-body effects on lepton pair production from lowmass enhancement and flow at RHIC and LHC energiesEur. Phys. J. C 71, 1760 (2011).S. Ghosh, Sourav Sarkar, Jan-e Alam, (VECC)
The $\rho$ meson in hot hadron matter and low mass dilepton spectra.Nucl. Phys. A 862, 294 (2011) S. Ghosh, Sourav Sarkar, Jan-e Alam, (VECC)
In-medium vector mesons and low mass lepton pairs from heavy ionCollisionsarXiv:1204.0893 [nucl-th] Sourav Sarkar, S. Ghosh
Elliptic flow of thermal dileptons as a probe of QCD matter Phys.Rev. C (R) 85 (2012) 031903
Payal Mohanty, Victor Roy, S. Ghosh, Santosh K. Das, Bedangadas Mohanty, Sourav Sarkar, Jane Alam, Asis K. Chaudhuri, (VECC)
Theoretical work on dilepton by VECC group (India)
Understanding low mass enhancement of SPS datain the language of Thermal Field Theory :
Mesonic collision rate
Baryonic collision rate
Bose enhancement of decay rate
Vacuum decay rate+
+
+
Hdronic Matter
Quark Matter
Total
+
(with Feynman boundary condition)
])()([2
),,( )( )( 0
ttitti ettetti
qtt
(with K.M.S.boundary condition)
])}()({
)}()([{2
),,(
)(
)(0
i
i
ef
efi
qD
Temporal Fourier transform
2200
1),(
mqqq
Temporal Fourier transform
220
210
120
110
00 ),(DD
DDqqDab
)(),,(][ 02
2
2
ttqttt
spatial Fourier transform spatial Fourier transform
)(),,(][ 02
2
2
cqD
)(),,(])[( 40
2 xxxxttmx
Free scalar propagator in vacuum satisfies
)(),,(])[( 40
2 xxxxDm cc
Free scalar propagator in medium satisfies
0,for
0,for ,),(),(
),(),(00
tDeCe
tBeAeDtiti
titi
Extra slides
Real part of self-energy
Baryonic loops
Mesonic loops